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If both the original combatants managed to whittle away each other's strength equally, then it might be that there would be 200 Starfleet and 800 Dominion ships left - in which case 600 Klingons might be needed. This would probably be true if all ships fought duels rather than group actions.

If they damaged each other in constant ratio, though, there could be 200 Starfleet but just 400 Dominion ships left, in which case just 200 Klingons would turn the tide. This would be true only if the individual ships of Starfleet were twice as good as the individual ships of the Dominion.

However, in melee battles like that, a combatant gaining a slight upper hand would typically be able to concentrate fire to quickly eliminate one opposing ship, leaving a greater ratio of ships to overpower the next victim, and again a greater ratio to deal with the next, and the next - until the initial underdog would be losing ships at an exponential rate, and the top dog would suffer virtually no losses. In which case 200 Starfleet ships left might indicate 1,100 Dominion ones left, and some 900 Klingons would be needed to make a difference.

It appears that the ships did not fight duels; that Starfleet ships in average indeed were better than the smallest Dominion ships, although probably inferior to the bigger ones; and that a melee did take place, but without chance of significant concentration of fire, of "destruction in detail" tactics or other such factors. So an arbitrary mixture of the above models can be postulated, and the number of Klingon ships chosen exactly as one pleases.

Well we only saw a small area of the battle, perhaps the outnumbered Federation fleet was holding it's own against the Dominion forces for the most part. The unexepected arrival of the Klingons might have caused some of the Dominion lines to crumble allowing the Federation Alliance to gain an advantage. And it would seem that just after the Dominion managed to clear the wormhole, the Dominion line collapsed as 200 Federation Allaince ships managed to break through and head for Deep Space Nine.

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On the continent of wild endeavour in the mountains of solace and solitude there stood the citadel of the time lords, the oldest and most mighty race in the universe looking down on the galaxies below sworn never to interfere only to watch.

It didn't look like there were even 100 when they arrived however I think that the Klingons had about 600 to evenly match the Starfleet numbers. In the episode before sacrifice Dukat stated he was pulling enough ships to crush them so I think Dukat saw how many ships were headed that way and pulled double the amount for them.

It was just the Klingons and the Federation at this point in the series, the Romulans weren't involved as of yet. But it was also the the Jem Hedar and the Cardasians. In the Next Generation we saw that at least Galaxies were able to top any Cardasian ship and the manner of talk was that at least in some ways Starfleet had superior technology to them. In DS9 we see the Klingons constantly pounding the Cardasians as well which was one of the reasons Dukat got the Dominion involved in the first place. I don't think to many Jem Hedar ships were taken out but a whole bunch Cardasian ships were. It would be interesting to know the ratio of Spoon Head ships to Jem Hedar ships that were sent in.

Also, I don't know much with the term "flanking". But I guess there is more damage to be done to a ship if you were able to hit broadside as opposed to straight on. There is simply more to hit and strike at. It looked as though the Klingons came in on a different vector than the Feds. When the came in the dominion were already immersed with the Federation. Maybe the element of surprise along with there different vector was enough to fully turn the tide.

In the episode before sacrifice Dukat stated he was pulling enough ships to crush them so I think Dukat saw how many ships were headed that way and pulled double the amount for them.

FWIW, conventional military wisdom has it that in 2D warfare involving "melee" weapons (be they swords, muskets or tank guns), you need to outnumber the defender 3:1 to triumph in attack. Nobody knows what would happen at 3D with "melee" weapons, but that's what Star Trek is all about - and geometry concerns would make it appear that the attacker is at a massive advantage from the very start, as the defender may be in the entirely wrong place, not knowing the direction from which the attack comes. So a ratio more like 1:6 might be sufficient for the attacker unless he gets drawn into a long attrition fight.

What Dukat needs to crush his enemy would depend on such considerations. A 2:1 advantage to the defender would be fine if Dukat were defending a point target and geometry did not matter - but Dukat tries to stop the invaders at a distance from the wormhole (the Defiant actually needs to go to warp to reach the wormhole after penetrating the Dominion line), so he's at the mercy of geometry all right, and might need to outnumber the Federation ten to one to stand any sort of a chance. If he only goes 2:1 and the Klingons suddenly come from another direction, he's at a massive disadvantage, even if the Klingon arrival only adjusts the ratio to something like 2:1.1.